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and Spherical, Cylindrical, and Ellipsoidal Harmonics, by William Elwood
Byerly
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Title: An Elementary Treatise on Fourier’s Series and Spherical,
Cylindrical, and Ellipsoidal Harmonics
With Applications to Problems in Mathematical Physics
Author: William Elwood Byerly
Release Date: August 19, 2009 [EBook #29779]
Language: English
Character set encoding: ISO-8859-1
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AN ELEMENTARY TREATISE
ON
FOURIER’S SERIES
AND
SPHERICAL, CYLINDRICAL, AND ELLIPSOIDAL
HARMONICS,
WITH
APPLICATIONS TO PROBLEMS IN MATHEMATICAL PHYSICS.
BY
WILLIAM ELWOOD BYERLY, Ph.D.,

instead of
δ
δx
for “partial derivative with respect to x.”
The course was at first, as I have said, an exposition of Riemann’s “Partielle
Differentialgleichungen.” In extending it, I drew largely from Ferrer’s “Spherical
Harmonics” and Heine’s “Kugelfunctionen,” and was somewhat indebted to
Todhunter (“Functions of Laplace, Bessel, and Lam´e”), Lord Rayleigh (“Theory
of Sound”), and Forsyth (“Differential Equations”).
In preparing the notes for publication, I have been greatly aided by the
criticisms and suggestions of my colleagues, Professor B. O. Peirce and Dr.
Maxime Bˆocher, and the latter has kindly contributed the brief historical sketch
contained in Chapter IX.
W. E. BYERLY.
Cambridge, Mass., Sept. 1893.
ii
ANALYTICAL TABLE OF CONTENTS.
CHAPTER I.
pages
Introduction 1–29
Art. 1. List of some important homogeneous linear partial differential equa-
tions of Physics.—Arts. 2–4. Distinction between the general solution and a
particular solution of a differential equation. Need of additional data to make
the solution of a differential equation determinate. Definition of linear and of
linear and homogeneous.—Arts. 5–6. Particular solutions of homogeneous lin-
ear differential equations may be combined into a more general solution. Need of
development in terms of normal forms.—Art. 7. Problem: Permanent state of
temperatures in a thin rectangular plate. Need of a development in sine series.
Example.—Art. 8. Problem: Transverse vibrations of a stretched elastic string.
A development in sine series suggested.—Art. 9. Problem: Potential function

tiples of the variable. Fourier’s series. Examples.—Art. 31. Extension of the
range within which the function and the series are equal. Examples.—Art. 32.
Fourier’s Integral obtained.
CHAPTER III.
Convergence of Fourier’s Series 56–69
Arts. 33–36. The question of the convergence of the sine series for unity
considered at length.—Arts. 37–38. Statement of the conditions which are
sufficient to warrant the development of a function into a Fourier’s series. His-
torical note. Art. 39. Graphical representation of successive approximations to
a sine series. Properties of a Fourier’s series inferred from the constructions.—
Arts. 40–42. Investigation of the conditions under which a Fourier’s series can
be differentiated term by term.—Art. 43. Conditions under which a function
can be expressed as a Fourier’s Integral.
CHAPTER IV.
Solution of Problems in Physics by the Aid of Fourier’s Inte-
grals and Fourier’s Series 70–135
Arts. 44–48. Logarithmic Potential. Flow of electricity in an infinite plane,
where the value of the Potential Function is given along an infinite straight line;
along two mutually perpendicular straight lines; along two parallel straight lines.
Examples. Use of Conjugate Functions. Sources and Sinks. Equipotential lines
and lines of Flow. Examples.—Arts. 49–52. One-dimensional flow of heat.
Flow of heat in an infinite solid; in a solid with one plane face at the tempera-
ture zero; in a solid with one plane face whose temperature is a function of the
time (Riemann’s solution); in a bar of small cross section from whose surface
heat escapes into air at temperature zero. Limiting state approached when the
temperature of the origin is a periodic function of the time. Examples.—Arts.
53–54. Temperatures due to instantaneous and to permanent heat sources
and sinks, and to heat doublets. Examples. Application to the case where
there is leakage.—Arts. 55–56. Transmission of a disturbance along an infinite
stretched elastic string. Examples.—Arts. 57–58. Stationary temperatures

Examples: Spheroidal conductors. Potential Function due to the attraction
of a material homogeneous circular disc. Examples: Homogeneous hemisphere;
Heterogeneous sphere; Homogeneous spheroids. Generalisation.—Art. 82. Leg-
endrian as a sum of cosines.—Arts. 83–84. Legendrian as the mth derivative
of the mth power of x
2
− −1.—Art. 85. Equations derivable from Legendre’s
Equation.—Art. 86. Legendrian as a Partial Derivative.—Art. 87. Legen-
drian as a Definite Integral. Arts. 88–90. Development in Zonal Harmonic
Series. Integral of the product of two Legendrians of different degrees. Integral
of the square of a Legendrian. Formulas for the coefficients of the series.—
Arts. 91–92. Integral of the product of two Legendrians obtained by the aid
of Legendre’s Equation; by the aid of Green’s Theorem. Additional formulas
for integration. Examples.—Arts. 93–94. Problems in Potential where the
value of the Potential Function is given on a spherical surface and has circular
TABLE OF CONTENTS v
symmetry about a diameter. Examples.—Art. 95. Development of a power
of x in Zonal Harmonic Series.—Art. 96. Useful formulas.—Art. 97. Devel-
opment of sin nθ and cos nθ in Zonal Harmonic Series. Examples. Graphical
representation of the first seven Surface Zonal Harmonics. Construction of suc-
cessive approximations to Zonal Harmonic Series. Arts. 98–99. Method of
dealing with problems in Potential when the density is given. Examples.—Art.
100. Surface Zonal Harmonics of the second kind. Examples: Conal Harmonics.
CHAPTER VI.
Spherical Harmonics 196–219
Arts. 101–102. Particular Solutions of Laplace’s Equation obtained. As-
sociated Functions. Tesseral Harmonics. Surface Spherical Harmonics. Solid
Spherical Harmonics. Table of Associated Functions. Examples.—Arts. 103–
108. Development in Spherical Harmonic Series. The integral of the product
of two Surface Spherical Harmonics of different degrees taken over the surface

from the base. Bessel’s Functions of a complex variable. Examples.—Art. 129.
Problem: Stationary temperatures in a cylinder when the temperatures of the
base are unsymmetrical. Bessel’s Functions of the nth order employed. Miscel-
laneous examples. Bessel’s Functions of fractional order.
CHAPTER VIII.
Laplace’s Equation in Curvilinear Co
¨
ordinates. Ellipsoidal
Harmonics 239–266
Arts. 130–131. Orthogonal Curvilinear Co¨ordinates in general. Laplace’s
Equation expressed in terms of orthogonal curvilinear co¨ordinates by the aid of
Green’s theorem.—Arts. 132–135. Spheroidal Co¨ordinates. Laplace’s Equation
in spheroidal co¨ordinates, in normal spheroidal co¨ordinates. Examples. Condi-
tion that a set of curvilinear co¨ordinates should be normal. Thermometric Pa-
rameters. Particular solutions of Laplace’s Equation in spheroidal co¨ordinates.
Spheroidal Harmonics. Examples. The Potential Function due to the attrac-
tion of an oblate spheroid. Solution for an external point. Examples.—Arts.
136–141. Ellipsoidal Co¨ordinates. Laplace’s Equation in ellipsoidal co¨ordinates.
Normal ellipsoidal co¨ordinates expressed as Elliptic Integrals. Particular solu-
tions of Laplace’s Equation. Lam´e’s Equation. Ellipsoidal Harmonics (Lam´e’s
Functions). Tables of Ellipsoidal Harmonics of the degrees 1, 2, and 3. Lam´e’s
Functions of the second kind. Examples. Development in Ellipsoidal Har-
monic series. Value of the Potential Function at any point in space when its
value is given at all points on the surface of an ellipsoid.—Art. 142. Conical
Co¨ordinates. The product of two Ellipsoidal Harmonics a Spherical Harmonic.—
Art. 143. Toroidal Co¨ordinates. Laplace’s Equation in toroidal co¨ordinates.
Particular solutions. Toroidal Harmonics. Potential Function for an anchor ring.
CHAPTER IX.
Historical Summary 267–274
APPENDIX.

[I]
where u represents the temperature at any point of the solid and t the time.
In the simplest case, that of a slab of infinite extent with parallel plane
faces, where the temperature can be regarded as a function of one co¨ordinate,
[I] reduces to
D
t
u = a
2
D
2
x
u, [II]
a form of considerable importance in the consideration of the problem of the
cooling of the earth’s crust.
In the problem of the permanent state of temperatures in a thin rectangular
plate, the equation [I] becomes
D
2
x
u + D
2
y
u = 0. [III]
In polar or spherical co¨ordinates [I] is less simple, it is
D
t
u =
a
2

D
t
(ru) = a
2
D
2
r
(ru). [V]
In cylindrical co¨ordinates [I] becomes
D
t
u = a
2
[D
2
r
u +
1
r
D
r
u +
1
r
2
D
2
φ
u + D
2

for the operation D
2
x
+ D
2
y
+ D
2
z
;
and with this notation equation [I] would be written D
t
u = a
2
∇
2
u.
INTRODUCTION. 2
for instance, in considering the transverse or the longitudinal vibrations of a
stretched elastic string, or the transmission of plane sound waves through the
air.
If in considering the transverse vibrations of a stretched string we take ac-
count of the resistance of the air [VIII] is replaced by
D
2
t
y + 2kD
t
y = a
2

D
r
z +
1
r
2
D
2
φ
z). [XI]
In the theory of Potential we constantly meet Laplace’s Equation
D
2
x
V +D
2
y
V + D
2
z
V = 0 [XII]
or ∇
2
V = 0
which in spherical co¨ordinates becomes
1
r
2

rD

1
r
2
D
2
φ
V + D
2
z
V = 0. [XIV]
In curvilinear co¨ordinates it is
h
1
h
2
h
3

D
ρ
1

h
1
h
2
h
3
D
ρ

3
V

= 0;
[XV]
where f
1
(x, y, z) = ρ
1
, f
2
(x, y, z) = ρ
2
, f
3
(x, y, z) = ρ
3
represent a set of surfaces which cut one another at right angles, no matter what
values are given to ρ
1
, ρ
2
, and ρ
3
; and where
h
2
1
= (D
x

2
+ (D
z
ρ
2
)
2
h
2
3
= (D
x
ρ
3
)
2
+ (D
y
ρ
3
)
2
+ (D
z
ρ
3
)
2
,
and, of course, must be expressed in terms of ρ

D
2
ρ
2
V + h
2
3
D
2
ρ
3
V = 0. [XVI]
INTRODUCTION. 3
2. A differential equation is an equation containing derivatives or differ-
entials with or without the primitive variables from which they are derived.
The general solution of a differential equation is the equation expressing the
most general relation between the primitive variables which is consistent with
the given differential equation and which does not involve differentials or deriva-
tives. A general solution will always contain arbitrary (i. e., undetermined)
constants or arbitrary functions.
A particular solution of a differential equation is a relation between the
primitive variables which is consistent with the given differential equation, but
which is less general than the general solution, although included in it.
Theoretically, every particular solution can be obtained from the general
solution by substituting in the general solution particular values for the arbitrary
constants or particular functions for the arbitrary functions; but in practice it is
often easy to obtain particular solutions directly from the differential equation
when it would be difficult or impossible to obtain the general solution.
3. If a problem requiring for its solution the solving of a differential equa-
tion is determinate, there must always be given in addition to the differential

satisfies the given differential equation, their sum will satisfy the equation; for if
the sum of the values in question is substituted in the equation each term of the
sum will give rise to a set of terms which must be equal to zero, and therefore
the sum of these sets must be zero.
6. It is generally possible to get by some simple device particular solutions
of such differential equations as those we have collected in Art. 1. The object of
the branch of mathematics with which we are about to deal is to find methods of
so combining these particular solutions as to satisfy any given conditions which
are consistent with the nature of the problem in question.
This often requires us to be able to develop any given function of the variables
which enter into the expression of these conditions in terms of normal forms
suited to the problem with which we happen to be dealing, and suggested by
the form of particular solution that we are able to obtain for the differential
equation.
These normal forms are frequently sines and cosines, but they are often
much more complicated functions known as Legendre’s Coefficients, or Zonal
Harmonics; Laplace’s Coefficients, or Spherical Harmonics: Bessel’s Functions,
or Cylindrical Harmonics; Lam´e’s Functions, or Ellipsoidal Harmonics, &c.
7. As an illustration, let us take Fourier’s problem of the permanent state
of temperatures in a thin rectangular plate of breadth π and of infinite length
whose faces are impervious to heat. We shall suppose that the two long edges of
the plate are kept at the constant temperature zero, that one of the short edges,
which we shall call the base of the plate, is kept at the temperature unity, and
that the temperatures of points in the plate decrease indefinitely as we recede
from the base; we shall attempt to find the temperature at any point of the
plate.
Let us take the base as the axis of X and one end of the base as the origin.
Then to solve the problem we are to find the temperature u of any point from
the equation
D

Hence u = e
αy±αxi 3
is a solution of [III], no matter what value may be given
to α.
This form is objectionable, since it involves an imaginary. We can, however,
readily improve it.
Take u = e
αy
e
αxi
, a solution of [III], and u = e
αy
e
−αxi
, another solution
of [III]; add these values of u and divide the sum by 2 and we have e
αy
cos αx.
(v. Int. Cal. Art. 35, [1].) Therefore by Art. 5
u = e
αy
cos αx (5)
is a solution of [III]. Take u = e
αy
e
αxi
and u = e
αy
e
−αxi

3
e
−3y
sin 3x + A
4
e
−4y
sin 4x + ··· (7)
A
1
, A
2
, A
3
, A
4
, &c., being undetermined constants.
When y = 0 (7) reduces to
u = A
1
sin x + A
2
sin 2x + A
3
sin 3x + A
4
sin 4x + ··· . (8)
If now it is possible to develop unity into a series of the form (8), our problem
is solved; we have only to substitute the coefficients of that series for A
1

for all values of x between 0 and π; hence our required solution is
u =
4
π

e
−y
sin x +
1
3
e
−3y
sin 3x +
1
5
e
−5y
sin 5x +
1
7
e
−7y
sin 7x + ···

(9)
for this satisfies the differential equation and all the given conditions.
If the given temperature of the base of the plate instead of being unity is a
function of x, we can solve the problem as before if we can express the given
function of x as a sum of terms of the form A sin mx, where m is a whole number.
The problem of finding the value of the potential function at any point of a

correct to the nearest degree. Ans. (a) 26
◦
; (b) 15
◦
; (c) 6
◦
.
8. As another illustration, we shall take the problem of the transverse
vibrations of a stretched string fastened at the ends, initially distorted into
some given curve and then allowed to swing.
Let the length of the string be l. Take the position of equilibrium of the
string as the axis of X, and one of the ends as the origin, and suppose the string
initially distorted into a curve whose equation y = f (x) is given.
We have then to find an expression for y which will be a solution of the
equation
D
2
t
y = a
2
D
2
x
y [VIII] Art. 1,
while satisfying the conditions
y = 0 when x = 0 (1)
y = 0 “ x = l (2)
y = f(x) “ t = 0 (3)
D
t

−(x±at)αi
, another solution of [VIII]. Add these
values of y and divide by 2 and we have cos α(x ±at). Subtract the second value
of y from the first and divide by 2i and we have sin α(x ± at).
y = cos α(x + at)
y = cos α(x −at)
y = sin α(x + at)
y = sin α(x −at)
are, then, solutions of [VIII]. Writing y successively equal to half the sum of the
first pair of values, half their difference, half the sum of the last pair of values,
and half their difference, we get the very convenient particular solutions of [VIII].
y = cos αx cos αat
y = sin αx sin αat
y = sin αx cos αat
y = cos αx sin αat.
If we take the third form
y = sin αx cos αat
it will satisfy conditions (1) and (4), no matter what value may be given to α,
and it will satisfy (2) if α =
mπ
l
where m is an integer.
If then we take
y = A
1
sin
πx
l
cos
πat

l
+ A
2
sin
2πx
l
+ A
3
sin
3πx
l
+ ··· (7)
If now it is possible to develop f(x) into a series of the form (7), we can solve
our problem completely. We have only to take the coefficients of this series as
values of A
1
, A
2
, A
3
··· in (6), and we shall have a solution of [VIII] which
satisfies all our given conditions.
In each of the preceding problems the normal function, in terms of which
a given function has to be expressed, is the sine of a simple multiple of the
variable. It would be easy to modify the problem so that the normal form
should be a cosine.
We shall now take a couple of problems which are much more complicated
and where the normal function is an unfamiliar one.
INTRODUCTION. 8
9. Let it be required to find the potential function due to a circular wire

sin
2
θ
D
2
φ
V = 0, [XIII] Art. 1,
subject to the condition
V =
M
(c
2
+ r
2
)
1
2
when θ = 0. (1)
From the symmetry of the ring, it is clear that the value of the potential
function must be independent of φ, so that [XIII] will reduce to
rD
2
r
(rV ) +
1
sin θ
D
θ
(sin θD
θ

dP
dθ

dθ
= 0, (3)
from which to obtain P .
Equation (3) can be simplified by changing the independent variable. Let
x = cos θ and (3) becomes
d
dx

(1 − x
2
)
dP
dx

+ m(m + 1)P = 0. (4)
5
See note on page 5.
INTRODUCTION. 9
Assume
6
now that P can be expressed as a sum or as a series of terms
involving whole powers of x multiplied by constant coefficients.
Let P =

a
n
x

= 0 (6)
and a
k+2
= −
m(m + 1) − k(k + 1)
(k + 1)(k + 2)
a
k
. (7)
If now any set of coefficients satisfying the relation (7) be taken, P =

a
k
x
k
will be a solution of (4).
If k = m, a
k+2
= 0, a
k+4
= 0, &c.
Since it will answer our purpose if we pick out the simplest set of coefficients
that will obey the condition (7), we can take a set including a
m
.
Let us rewrite (7) in the form
a
k
= −
(k + 2)(k + 1)

1
.
P = a
m

x
m
−
m(m − 1)
2.(2m − 1)
x
m−2
+
m(m − 1)(m − 2)(m − 3)
2.4.(2m − 1)(2m − 3)
x
m−4
− ···

where a
m
is entirely arbitrary, is, then, a solution of (4). It is found convenient
to take a
m
equal to
(2m − 1)(2m − 3) ···1
m!
and it can be shown that with this value of a
m
P = 1 when x = 1.

− ···

. (9)
We have:
P
0
(x) = 1 or P
0
(cos θ) = 1
P
1
(x) = x “ P
1
(cos θ) = cos θ
P
2
(x) =
1
2
(3x
2
− 1) “ P
2
(cos θ) =
1
2
(3 cos
2
θ − 1)
P

4
θ − 30 cos
2
θ + 3)
P
5
(x) =
1
8
(63x
5
− 70x
3
+ 15x) or
P
5
(cos θ) =
1
8
(63 cos
5
θ − 70 cos
3
θ + 15 cos θ).







m
(cos θ) as a particular solution of (3). P
m
(x) or P
m
(cos θ) is a new function,
known as a Legendre’s Coefficient, or as a Surface Zonal Harmonic, and occurs
as a normal form in many important problems.
V = r
m
P
m
(cos θ) is a particular solution of (2) and r
m
P
m
(cos θ) is sometimes
called a Solid Zonal Harmonic.
We can now proceed to the solution of our original problem.
V = A
0
r
0
P
0
(cos θ) + A
1
rP
1
(cos θ) + A

r
3
+ ··· ,
since, as we have said, P
m
(x) = 1 when x = 1, or P
m
(cos θ) = 1 when θ = 0.
By our condition (1)
V =
M
(c
2
+ r
2
)
1
2
when θ = 0.
By the Binomial Theorem
M
(c
2
+ r
2
)
1
2
=
M


P
0
(cos θ) −
1
2
r
2
c
2
P
2
(cos θ) +
1.3
2.4
r
4
c
4
P
4
(cos θ)
−
1.3.5
2.4.6
r
6
c
6
P


r = .6, θ =
π
2

;
(c)

r = .2, θ =
π
2

;
Ans. (a) .98; (b) .99; (c) 1.01; (d) .86;
(e) .90; (f ) 1.00; (g) 1.10.
The unit used is the potential due to a pound of mass concentrated at a point
and attracting a second pound of mass concentrated at a point, the two points
being a foot apart.
10. A slightly different problem calling for development in terms of Zonal
Harmonics is the following:
Required the permanent temperatures within a solid sphere of radius 1, one
half of the surface being kept at the constant temperature zero, and the other
half at the constant temperature unity.
Let us take the diameter perpendicular to the plane separating the unequally
heated surfaces as our axis and let us use spherical co¨ordinates. As in the last
problem, we must solve the equation
rD
2
r
(ru) +

π
2
and u = 0 from θ =
π
2
to θ = π, (2)
INTRODUCTION. 12
when r = 1.
As we have seen u = r
m
P
m
(cos θ) is a particular solution of (1), m being
any positive whole number, and
u = A
0
r
0
P
0
(cos θ) + A
1
rP
1
(cos θ) + A
2
r
2
P
2

3
P
3
(cos θ) + ··· (4)
If then we can develop our function of θ which enters into equation (2) in a
series of the form (4), we have only to take the coefficients of that series as the
values of A
0
, A
1
, A
2
, &c., in (3) and we shall have our required solution.
11. As a last example we shall take the problem of the vibration of
a stretched circular membrane fastened at the circumference, that is, of an
ordinary drumhead. We shall suppose the membrane initially distorted into
any given form which has circular symmetry
7
about an axis through the centre
perpendicular to the plane of the boundary, and then allowed to vibrate.
Here we have to solve
D
2
t
z = c
2

D
2
r

2
r
z +
1
r
D
r
z

(4)
and this is the equation for which we wish to find a particular solution.
We shall employ a device not unlike that used in Art. 9.
Assume
8
z = R.T where R is a function of r alone and T is a function of t
alone. Substitute this value of z in (4) and we get
RD
2
t
T = c
2
T

D
2
r
R +
1
r
D

1
r
dR
dr

. (5)
The second member of (5) does not involve t, therefore its equal the first member
must be independent of t. The first member of (5) does not involve r, and
consequently since it contains neither t nor r, it must be constant. Let it equal
−µ
2
, where µ of course is an undetermined constant.
Then (5) breaks up into the two differential equations
d
2
T
dt
2
+ µ
2
c
2
T = 0 (6)
d
2
R
dr
2
+
1

n
and substitute in (8). We get

[n(n − 1)a
n
x
n−2
+ na
n
x
n−2
+ a
n
x
n
] = 0,
an equation which must be true no matter what the value of x. The coefficient
of any given power of x, as x
k−2
, must, then, vanish, and
k(k − 1)a
k
+ ka
k
+ a
k−2
= 0
or k
2
a

2
.
Then a
2
= −
a
0
2
2
a
4
=
a
0
2
2
.4
2
a
6
= −
a
0
2
2
.4
2
.6
2
, &c.

convergent.
Take a
0
= 1, and then R = J
0
(x) where
J
0
(x) = 1 −
x
2
2
2
+
x
4
2
2
.4
2
−
x
6
2
2
.4
2
.6
2
+

+
r
6
2
2
.4
2
.6
2
+ ··· ,
where r is the modulus of x, is convergent for all values of r. For the ratio of
the n + 1st term of this series to the nth term is
r
2
4n
2
and approaches zero as
its limit as n is indefinitely increased, no matter what the value of r. Therefore
J
0
(x) is absolutely convergent.
J
0
(x) is a new and important form. It is called a Bessel’s Function of the
zeroth order, or a Cylindrical Harmonic.
Equation (8) was obtained from (7) by the substitution of x = µr, therefore
R = J
0
(µr) = 1 −
(µr)

J
0
(µa) = 0; (11)
that is, µ must be a root of (11) regarded as an equation in µ.
It can be shown that J
0
(x) = 0 has an infinite number of real positive roots,
any one of which can be obtained to any required degree of approximation
without serious difficulty. Let x
1
, x
2
, x
3
, ··· be these roots. Then if
x
1
a
= µ
1
,
x
2
a
= µ
2
,
x
3
a

, A
2
, A
3
, &c., are any constants, is a solution of (4) which satisfies
conditions (2) and (3).
When t = 0 (12) reduces to
z = A
1
J
0
(µ
1
r) + A
2
J
0
(µ
2
r) + A
3
J
0
(µ
3
r) + ··· . (13)
If then f(r) can be expressed as a series of the form just given, the solution of
our problem can be obtained by substituting the coefficients of that series for
A
1

2
t
J
0
(µ
2
r) + A
3
e
−a
2
µ
2
3
t
J
0
(µ
3
r) + ···
where µ
1
, µ
2
, &c., are the roots of J
0
(µc) = 0, and where
1 = A
1
J

If these equations are familiar ones their solutions can be written down at
once; if unfamiliar, the device used in problems 3 and 5 is often serviceable,
namely, that of assuming that the dependent variable can be expressed as a
sum or series of terms involving whole powers of the independent variable, and
then determining the coefficients.
Let us consider again the equations used in the first, second and third prob-
lems.
(a) D
2
x
u + D
2
y
u = 0 (1)
Assume u = X.Y where X involves x but not y, and Y involves y but not x.
Substitute in (1), Y D
2
x
X + XD
2
y
Y = 0,
or, since we are now dealing with functions of a single variable,
1
X
d
2
X
dx
2

2
.
Then
d
2
Y
dy
2
− α
2
Y = 0 (3)
and
d
2
X
dx
2
+ α
2
X = 0; (4)
and if (3) and (4) can be solved, we can solve (1). They have for their complete
solutions
Y = Ae
αy
+ Be
−αy
and X = C sin αx + D cos αx. (v. Int. Cal. p. 319, § 21.)
Hence Y = e
αy
and Y = e

d
2
T
dt
2
=
1
X
d
2
X
dx
2
; (2)
hence as in the last case
1
X
d
2
X
dx
2
is a constant; call it −α
2
, and (2) breaks up
into
d
2
X
dx